{
 "what": "The MacMahon minimum-universal-sets claim (Jason Davies, 2026-11-01), the SUFFICIENCY side decided from the released witnesses with our own geometry (instruments/macmahon/check.py: our 24 proper rotations from signed axis permutations of determinant +1, our eight corner slots, a perfect matching of cubes to slots searched here; the claimant's rotation indices and slot order ignored): T1 the claimant's label table lists all 30 varieties once; T2 the ten witnessed sets are the ten two-triangle sets; T3 all 300 witnessed constructions build their target. NOT decided here: that no other 12-multiset and no 11-multiset is universal (the necessity half).",
 "claim": "exactly ten labelled 12-cube collections of MacMahon cubes are universal (2×2×2 models of all 30 varieties, exteriors only, repeats allowed), and they are the two-triangle sets of Conway's tableau — Johnson–Roldán Conjectures 4 and 8",
 "source": {
  "repo": "https://github.com/jasondavies/macmahon-min-universal-sets",
  "commit": "7561f6d63b64f705824d05443050d0b8f532f2e3",
  "registered_commit": "d8856c97e43e6fa66383c81b8ac4f5bd1c285bc8",
  "pins": {
   "results/universal_witnesses.json": "18ded406ca9a70f2a8a6ca0bf006433a7cff691ea912d3f4af9cc1b0b67e341a",
   "results/geometry.json": "a17aed1869f73ed87e2da363421925e33b52cd6753d527a239614cbc0b8534c2",
   "results/verification.json": "c1cbf21bb7504333ecf2ba4eea19d6b110bbd45add4bb68fed86e76c46af43d5",
   "results/extremal_witnesses.json": "2ca687ad17bf7a45284f23a8ec35d7c2a8580cb9d15a009976e6b5ffa26f0512",
   "MacMahon/Data.lean": "51c72217cbdc75f2e2daea4a45a457d688c4453dcea3300bd22214d462d3b12b",
   "MacMahon/Definitions.lean": "e809de0d8f77b7d0a9607f6f4105392b82f5c47b599807d26464cb7921dfaa77",
   "MacMahon/Theorems.lean": "48d1ad91857e6f45653317d7ae309ef4594ac735161966400890a3b1439c2c75",
   "MacMahon/Types.lean": "658cd83486c25387df5c226f967c6cd89499d1290d77d95b233e3a7d8d573d39",
   "Challenge.lean": "7bfd58397db1135e0ff5233e94792fa1da3dcbec27a2b219f1e284f309c0c313",
   "formalization.yaml": "2c49e2fb77993f2c5c004cefcf28985ba50194903cd39690ef059b79f2046cc5",
   "README.md": "f3521cee5f75982e8fa9ce8e9984c475dd570b929f8bbe4c04af4a2c3c109dd4",
   "lean-toolchain": "ca5f1f601d34c49079e38d7e7041eb0f2a9c227fbc1df0f404820069a052330c",
   "lakefile.toml": "6a471ffa9e7fc64d01d71f502b93d6d38d68a5fa2ec810de611798bc36f08a69",
   "lake-manifest.json": "108b1bae4eaa7244e16190e8d9085d78f4a1fb90b0d1950a57b6f84697ff38c9",
   "paper/macmahon-min-universal-sets.tex": "ee059bbddbbc85504dbfa324b5ebad96761f91238c3befe0d85d02dfdbd5bf03",
   "scripts/verify_cubes.py": "8296f3fe019b51a6fce501ddf3f6e1883f1dbd6710b55c988d674abbfee728f2",
   "src/cubes.cpp": "821f99834d987d47a47cfd900c22bab81eb92c62edfc7eceda74726012eaf975",
   "results/experiments.jsonl": "a6d6cbe9e73ed47fb4eb1a7c7e0118db50b09d44b8251f55f70e76bfcc9dbbfb",
   "comparator.json": "334c76af192126fffe1cba3df7dfeac1781ef4117c2d55d72ba15c8c790ad7f4",
   "Solution.lean": "9eb99bff2e12cb3658257c74d312a2e3728d859126d001467553b567bc9f890c",
   "MacMahon.lean": "baa4b5fba2419a8dc907165a7af2db67ff1b999d5b71cf87e00db479785c601f",
   "validation/Axioms.lean": "41a76460151c3969f52c92f6690edb006049d1f0408c715a2caa0f3428293ca1",
   "validation/axioms.txt": "110e1e4da52dfde6ffae47ca45dff8abe5bd6c47bf1c287a45cfcdf6d34d3635"
  }
 },
 "generated": "2026-11-03T23:09:39Z",
 "seconds": 0.3,
 "decided": {
  "sufficiency": "CERTIFIED",
  "necessity": "NOT DECIDED HERE (the Lean proof and the claimant's search; a later stage)"
 },
 "verdict": "CERTIFIED",
 "facts": {
  "T1": {
   "labels": 30,
   "six_distinct_colours_each": true,
   "rotation_classes": 30,
   "classes_of_all_720_colourings": 30,
   "verdict": "CERTIFIED"
  },
  "T2": {
   "sets": 10,
   "distinct_sets": 10,
   "two_triangle_sets": 10,
   "partitions_of_six_into_two_triples": 10,
   "verdict": "CERTIFIED"
  },
  "T3": {
   "constructions": 300,
   "built": 300,
   "failed": [],
   "verdict": "CERTIFIED"
  }
 },
 "red_controls": [
  {
   "red": "a witness with one cube swapped for a cube outside the set is NOT built",
   "fired": true
  },
  {
   "red": "the same eight cubes against the target with two faces exchanged are NOT built",
   "fired": true
  },
  {
   "red": "a rotation table that admits reflections (48 maps) merges the 30 varieties into 15, so T1 would fail under it",
   "fired": true
  }
 ],
 "failed": [],
 "reds_dead": []
}